1. **Problem statement:** We have monthly income data for 15 households grouped into income classes with given histogram bar heights (density). We need to create a frequency table, calculate mean and standard deviation. 2. **Step 1: Calculate frequencies from histogram densities and class widths.** Frequency = density \times class width \times total number of households Given classes and densities: - 20-30: width = 10, density = 0.1 - 30-40: width = 10, density = 0.3 - 40-50: width = 10, density = 0.4 - 50-90: width = 40, density = 0.15 - 90-110: width = 20, density = 0.05 Calculate frequencies: - 20-30: $0.1 \times 10 \times 15 = 15$ - 30-40: $0.3 \times 10 \times 15 = 45$ - 40-50: $0.4 \times 10 \times 15 = 60$ - 50-90: $0.15 \times 40 \times 15 = 90$ - 90-110: $0.05 \times 20 \times 15 = 15$ Sum of frequencies = $15 + 45 + 60 + 90 + 15 = 225$ which is inconsistent with total 15 households, so we must interpret densities as relative frequencies per class width. 3. **Step 2: Calculate relative frequencies per class:** Relative frequency per class = density \times class width - 20-30: $0.1 \times 10 = 1.0$ - 30-40: $0.3 \times 10 = 3.0$ - 40-50: $0.4 \times 10 = 4.0$ - 50-90: $0.15 \times 40 = 6.0$ - 90-110: $0.05 \times 20 = 1.0$ Sum = $1 + 3 + 4 + 6 + 1 = 15$ matches total households. 4. **Step 3: Frequency table:** | Class | Width | Frequency | |--------|-------|-----------| | 20-30 | 10 | 1 | | 30-40 | 10 | 3 | | 40-50 | 10 | 4 | | 50-90 | 40 | 6 | | 90-110 | 20 | 1 | 5. **Step 4: Calculate class midpoints:** - 20-30: 25 - 30-40: 35 - 40-50: 45 - 50-90: 70 - 90-110: 100 6. **Step 5: Calculate mean:** $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{1\times25 + 3\times35 + 4\times45 + 6\times70 + 1\times100}{15} = \frac{25 + 105 + 180 + 420 + 100}{15} = \frac{830}{15} = 55.33$$ 7. **Step 6: Calculate variance:** $$s^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$$ Calculate each term: - $(25 - 55.33)^2 = 915.11$, contribution: $1 \times 915.11 = 915.11$ - $(35 - 55.33)^2 = 413.44$, contribution: $3 \times 413.44 = 1240.32$ - $(45 - 55.33)^2 = 106.78$, contribution: $4 \times 106.78 = 427.11$ - $(70 - 55.33)^2 = 213.44$, contribution: $6 \times 213.44 = 1280.64$ - $(100 - 55.33)^2 = 1980.44$, contribution: $1 \times 1980.44 = 1980.44$ Sum contributions: $915.11 + 1240.32 + 427.11 + 1280.64 + 1980.44 = 5843.62$ Variance: $$s^2 = \frac{5843.62}{15} = 389.57$$ Standard deviation: $$s = \sqrt{389.57} = 19.74$$ **Final answers:** - Mean monthly income: $55.33$ thousand kroner - Standard deviation: $19.74$ thousand kroner